{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article10687","name":"Surjective H-Colouring over Reflexive Digraphs","abstract":"The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs. \r\n\r\nChen [2014] proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms are essentially unary. We give the first concrete application of his result by showing that every endo-trivial reflexive digraph H has this property. We then use the concept of endo-triviality to prove, as our main result, a dichotomy for Surjective H-Colouring when H is a reflexive tournament: if H is transitive, then Surjective H-Colouring is in NL, otherwise it is NP-complete.\r\n\r\nBy combining this result with some known and new results we obtain a complexity classification for Surjective H-Colouring when H is a partially reflexive digraph of size at most 3.","keywords":["Surjective H-Coloring","Computational Complexity","Algorithmic Graph Theory","Universal Algebra","Constraint Satisfaction"],"author":[{"@type":"Person","name":"Larose, Benoit","givenName":"Benoit","familyName":"Larose"},{"@type":"Person","name":"Martin, Barnaby","givenName":"Barnaby","familyName":"Martin"},{"@type":"Person","name":"Paulusma, Daniel","givenName":"Daniel","familyName":"Paulusma"}],"position":49,"pageStart":"49:1","pageEnd":"49:14","dateCreated":"2018-02-27","datePublished":"2018-02-27","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Larose, Benoit","givenName":"Benoit","familyName":"Larose"},{"@type":"Person","name":"Martin, Barnaby","givenName":"Barnaby","familyName":"Martin"},{"@type":"Person","name":"Paulusma, Daniel","givenName":"Daniel","familyName":"Paulusma"}],"copyrightYear":"2018","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2018.49","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6299","volumeNumber":96,"name":"35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)","dateCreated":"2018-02-27","datePublished":"2018-02-27","editor":[{"@type":"Person","name":"Niedermeier, Rolf","givenName":"Rolf","familyName":"Niedermeier"},{"@type":"Person","name":"Vall\u00e9e, Brigitte","givenName":"Brigitte","familyName":"Vall\u00e9e"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article10687","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6299"}}}